U find the range of f ( x ) = − 5 x 2 + 1 x + 5 ?

Question

U find the range of #f(x)=−5x^2+1x+5?#

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Answer 1 · 2017-05-19T14:26:16

The Range is #{ y in RR : y le 5.05}=(-oo,5.05].#

I hope it is intended to find the Range of the function,

#f(x)=-5x^2+x+5, x in RR.#

#f(x)=-5(x^2-1/5x-1)#

Completing the Square, we have,

#f(x)=-5{(x^2-1/5x+1/100)-101/100},#

#=-5{(x-1/10)^2-101/100},#

#:. f(x)=-5(x-1/10)^2+505/100.#

#because AA x in RR, (x-1/10)^2 ge 0, -5(x-1/10)^2 le 0,#

#:. 505/100-5(x-1/10)^2 le 505/100, i.e., AA x in RR, f(x) le 505/100.#

Hence, the Range is #{ y in RR : y le 5.05}=(-oo,5.05].#

Enjoy Maths.!